W. Barth, "Using Special Arithmetic in Geometric Algorithms", to appear in Numerical Methods and Error-Bounds, Proceedings of the IMACS-GAMM International Symposium, University of Oldenburg, Germany, July 1995, Akademie Verlag, Berlin, 1996.
The application of interval arithmetic shown in this paper deals with geometric algorithms. The purpose for which we use this reliable computation is different from the usual one. It is not to find an inclusion for the final results, but it is to control the algorithm correctly, i.e., to find the right branch the algorithm has to follow from the signs of intermediate results, this means assuring reliable control. The problem arises when working with the method of Edelsbrunner and M\"ucke for eliminating all special cases in geometric algorithms. Of course the sign of an intermediate result (arithmetic expression) can only be determined correctly without any exception by exact arithmetic. But it is much more efficient to calculate such an expression first with interval arithmetic, and only in the few cases where the resulting interval contains zero the time consuming exact calculation has to be performed.